R f f S TCSf f 1 tan2 P24

which is identical to the value for the central ray, as the geometry of the parabola requires. Depending on the simplifications and approximations we allow in the derivation, we find slightly different representations for the path difference. We compare these now.

J lX

Fig. 2.6. Illustration of the geometry of the axial defocus

i). Assuming that cos y changes insignificantly with the focus shift, we find for the defocused ray p' ° 2 (f + d)/(1 + cos y), leading to a path difference of p' - p ° 2 d /(1 + cos y).

Note that upon reflection both rays travel to the aperture of the antenna essentially parallel and we ignore the small length difference over that path. The phase error over the aperture is proportional to the difference in the pathlength change between any ray to a radius r and the central ray. Using several of the earlier geometric formulations for the paraboloid, we find for the pathlength difference Aa over the aperture, caused by an axial defocus d

Da r = d (r4sy - l) = d l-COSf = d tan2 (f) = d (Jf )2 (2.25)

We see that, subject to the assumption that cos y changes insignificantly between the focused and defocused situation, the phase error due to axial defocusing is a simple quadratic function in the radial aperture coordinate r. The maximum error at the edge of the aperture with diameter d is thus

ii). As an alternative, better approximation we can apply the cosine-rule to the triangle FPF'. We have p,2 = p2 + d2 - 2 pdcos(p-y) = p2 + d2 + 2 pd cos y, from which we obtain the path length change of the ray to P

2 pd cos y + d2 d2 p, - p =----pr+-p---° d cos y + -2-p ° d cos y (2.27)

and hence the path error over the aperture is given by

0 0

Post a comment